Some Extensions of Thomassen's Theorem to Longer Paths

Let $G$ be a planar embedding with list-assignment $L$ and outer cycle $C$, and let $P$ be a path of length at most four on $C$, where each vertex of $G\setminus C$ has a list of size at least five and each vertex of $C\setminus P$ has a list of size at least three. In this paper, we prove some results about partial $L$-colorings $\phi$ of $C$ with the property that any extension of $\phi$ to an $L$-coloring of $\text{dom}(\phi )\cup V(P)$ extends to $L$-color all of $G$. We use these results in a later sequence of papers to prove some results about list-colorings of high-representativity embeddings on surfaces.